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Let $Q$ be a loop such that $\left|Q\right|$ is square-free and the inner mapping group $I\left(Q\right)$ is nilpotent. We show that $Q$ is centrally nilpotent of class at most two.

Let $Q$ be a finite commutative loop and let the inner mapping group $I\left(Q\right)\cong C_{p^n}\times C_{p^n}$, where $p$ is an odd prime number and $n\ge 1$. We show that $Q$ is centrally nilpotent of class two.

Let $G$ be a finite group with a dicyclic subgroup $H$. We show that if there exist $H$-connected transversals in $G$, then $G$ is a solvable group. We apply this result to loop theory and show that if the inner mapping group $I\left(Q\right)$ of a finite loop $Q$ is dicyclic, then $Q$ is a solvable loop. We also discuss a more general solvability criterion in the case where $I\left(Q\right)$ is a certain type of a direct product.